Solving the Kinetic Ising Model with Non-Reciprocity

TL;DR

This paper introduces an exactly solvable non-reciprocal kinetic Ising model in one dimension, revealing novel non-equilibrium phenomena, phase regimes, and scaling behaviors induced by non-reciprocity.

Contribution

It provides the first exact solutions for a non-reciprocal kinetic Ising model, exploring its dynamical phases, boundary effects, and low-energy scaling properties.

Findings

Identification of non-reciprocity induced frustration and wave phenomena

Discovery of phase regimes separated by Nth-order exceptional points

Demonstration that long-time order exists only at zero temperature

Abstract

Non-reciprocal interactions are a generic feature of non-equilibrium systems. We define a non-reciprocal generalization of the kinetic Ising model in one spatial dimension. We solve the model exactly using two different approaches for infinite, semi-infinite and finite systems with either periodic or open boundary conditions. The exact solution allows us to explore a range of novel phenomena tied to non-reciprocity like non-reciprocity induced frustration and wave phenomena with interesting parity-dependence for finite systems of size $N$. We study dynamical questions like the approach to equilibrium with various boundary conditions. We find new regimes, separated by $N^{th}$-order exceptional points, which can be classified as overdamped, underdamped and critically damped phases. Despite these new regimes, long-time order is only present at zero temperature. Additionally, we explore the low-energy behavior of the system in various limits, including the ageing and spatio-temporal Porod regimes, demonstrating that non-reciprocity induces unique scaling behavior at zero temperature. Lastly, we present general results for systems where spins interact with no more than two spins, outlining the conditions under which long-time order may exist.